Question 11.SP.11: A projectile is fired with an initial velocity of 800 ft/s a...
A projectile is fired with an initial velocity of 800 ft/s at a target B located 2000 ft above the gun A and at a horizontal distance of 12,000 ft. Neglect-ing air resistance, determine the value of the firing angle α needed to hit the target.
STRATEGY: This is a projectile motion problem, so you can consider the vertical and horizontal motions separately. First determine the equations gov-erning the motion in each direction, and then use them to find the firing angle.
MODELING and ANALYSIS:
Horizontal Motion. Place the origin of the coordinate axes at the gun (Fig. 1). Then
Substituting into the equation of uniform horizontal motion, you obtain
x=\left(v_{x}\right)_{0} t \quad x=(800 \cos \alpha) tObtain the time required for the projectile to move through a horizontal distance of 12,000 ft by setting x equal to 12,000 ft.
\begin{aligned}12,000 &=(800 \cos \alpha) t \\t &=\frac{12,000}{800 \cos \alpha}=\frac{15}{\cos \alpha}\end{aligned}Vertical Motion. Again, place the origin at the gun (Fig. 2).
\left(v_{y}\right)_{0}=800 \sin \alpha \quad a=-32.2 \mathrm{ft} / \mathrm{s}^{2}Substituting into the equation for constant acceleration in the vertical direction, you obtain
y=\left(v_{y}\right)_{0} t+\frac{1}{2} a t^{2} \quad y=(800 \sin \alpha) t-16.1 t^{2}Projectile Hits Target. When x = 12,000 ft, you want y = 2000 ft. Substituting for y and setting t equal to the value found previously, you have
\begin{gathered}2000=800 \sin \alpha \frac{15}{\cos \alpha}-16.1\left(\frac{15}{\cos \alpha}\right)^{2} \\Since 1 / \cos ^{2} \alpha=\sec ^{2} \alpha=1+\tan ^{2} \alpha, \text { you have } \\2000=800(15) \tan \alpha 16.1\left(15^{2}\right)\left(1+\tan ^{2} \alpha\right) \\3622 \tan ^{2} \alpha-12,000 \tan \alpha+5622=0\end{gathered}Solving this quadratic equation for tan α gives you
tan α = 0.565 and tan α =2.75
α = 29.5° and α = 70.0°
The target will be hit if either of these two firing angles is used (Fig. 3).
REFLECT and THINK: It is a well-known characteristic of projectile motion that you can hit the same target by using either of two firing angles. We used trigonometry to write the equation in terms of tan α, but most calculators or computer programs like Maple, Matlab, or Mathematica also can be used to solve (1) for α. You must be careful when using these tools, however, to make sure that you find both angles.
